STA: "IP-PosDefinite.3x3.txt"	Tuesday, 14Nov2023

      Given colvecs b0, b1 and b2 in |R^3  [the vectors
    are defined below], we compute the unique
    positive-definite matrix Q such that
    the inner-product  (acting on 3x1 colvecs)

        (c0, c1) |->   TPose(c0) * Q * c1

    makes {b0,b1,b2} an ortho-normal basis of |R^2.

================================================================

                LISP CODE
(setq b0 (mat-make-colvec 1  0  1)
      b1 (mat-make-colvec 0  2  1)
      b2 (mat-make-colvec 1 -1  0) )


;; The following fnc will realize B-orthonormal IP, 
;; once we have computed positive-definite matrix Q.
(defun IP-by-Q (ColX ColY) 
               (mat-E (mat-mul (mat-tpose ColX) Q ColY) 0 0) )


;; Computing 2x2 matrix Q:
(setq Id-ToE-FrB (mat-Horiz-concat b0 b1 b2))
    [  1   0   1 ]
    [  0   2  -1 ]
    [  1   1   0 ]

(setq Id-ToB-FrE (mat-inverse? Id-ToE-FrB))
    [ -1  -1   2 ]
    [  1   1  -1 ]
    [  2   1  -2 ]


;; Let's check that we have the correct CoB matrix.
                              [ 1 ]
(mat-mul Id-ToB-FrE  b0)  =>  [ 0 ]
                              [ 0 ]
                              
                              [ 0 ]
(mat-mul Id-ToB-FrE  b1)  =>  [ 1 ]
                              [ 0 ]
                              
                              [ 0 ]
(mat-mul Id-ToB-FrE  b2)  =>  [ 0 ]
                              [ 1 ]
                              
(setq Q (mat-mul (mat-tpose Id-ToB-FrE) Id-ToB-FrE))
    [  6   4  -7 ]
    [  4   3  -5 ]
    [ -7  -5   9 ]

;;; Checking Checking Checking
(list (IP-by-Q b0 b1) (IP-by-Q b0 b2) (IP-by-Q b1 b2))
    (0, 0, 0)
;; Cool: Vectors b0,b1,b2 are pairwise-orthogonal


(list (IP-by-Q b0 b0) (IP-by-Q b1 b1) (IP-by-Q b2 b2))
    (1, 1, 1)
;; And each is of length 1.
;; [Technically, we view list (1, 1, 1) as (1^2, 1^2, 1^2).]

;; The computation was Smooth Sailing...
END: "IP-PosDefinite.3x3.txt"